Definition:Preimage/Relation/Relation
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Definition
Let $\RR \subseteq S \times T$ be a relation.
Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$, defined as:
- $\RR^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \RR}$
The preimage of $\RR \subseteq S \times T$ is:
- $\Preimg \RR := \RR^{-1} \sqbrk T = \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$
Also known as
Some sources, for example 1975: T.S. Blyth: Set Theory and Abstract Algebra, call this the domain of $\RR$.
However, this term is discouraged, as it is also seen used to mean the entire set $S$, including elements of that set which have no images.
Also see
Technical Note
The $\LaTeX$ code for \(\Preimg {f}\) is \Preimg {f}
.
When the argument is a single character, it is usual to omit the braces:
\Preimg f
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings